MULTISCALE LOCALIZED DIFFERENTIAL QUADRATURE IN 2D...

MULTISCALE LOCALIZED DIFFERENTIAL QUADRATURE IN 2D

PARTIAL DIFFERENTIAL EQUATION FOR MECHANICS OF SHAPE

MEMORY ALLOYS

CHEONG HUI TING

UNIVERSITI TEKNOLOGI MALAYSIA

MULTISCALE LOCALIZED DIFFERENTIAL QUADRATURE IN 2D PARTIAL

DIFFERENTIAL EQUATION FOR MECHANICS OF SHAPE MEMORY

ALLOYS

CHEONG HUI TING

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

JUNE 2017

iii

To my beloved family, for your love and support.

To my friends, for your wit, intelligence and guidance in life.

iv

ACKNOWLEDGEMENT

First and foremost, I would like to express my sincere thanks to my

supervisor, Dr. Yeak Su Hoe who greatly assisted in the preparation of this research.

His efforts in patiently guiding, supporting and giving constructive suggestions are

very much appreciated.

My thanks are due to my family members for their unique perspectives and

support. Without their support, this project would have been difficult to solve.

Finally, all my course mates and friends deserve thanks for encouragement and

suggestion for the improvements they gave me.

v

ABSTRACT

In this research, the applicability of the Multiscale Localized Differential

Quadrature (MLDQ) method in two-dimensional shape memory alloy (SMA) model

was explored. The MLDQ method was governed in solving several partial

differential equations. Besides, the finite difference (FD) method was used to solve

some examples of partial differential equations and the solutions obtained were

compared with those obtained by MLDQ method in order to show the accuracy of

the numerical method. The MLDQ method was developed by increasing the number

of grid points in critical region, and approximating the derivatives at the certain

selected grid points. This present method together with the fourth-order Runge-Kutta

(RK) method has been applied in differential equations such as wave equation and

high gradient problems,. The MLDQ method can achieves accurate numerical

solutions compared with FD method which is a low order numerical method by using

a few number of grid points. The multiscale method was employed at the critical

region which can break down the region of interest from coarser into finer grid points.

Furthermore, FORTRAN programs were developed based on MLDQ method in

solving some problems as above. The shared memory architecture of parallel

computing was done by using OpenMP in order to reduce the time taken in

simulating the numerical results. Consequently, the results show that the MLDQ

method was a good numerical technique in two-dimensional SMA.

vi

ABSTRAK

Dalam kajian ini, kesesuaian kaedah Multiscale Berbeza Kuadratur Setempat

(MLDQ) dalam model dua dimensi Aloi Memori Bentuk (SMA) telah diterokai.

Kaedah MLDQ telah dibangunkan dalam menyelesaikan beberapa persamaan

pembezaan separa. Selain itu, kaedah Perbezaan Terhingga (FD) telah digunakan

untuk menyelesaikan beberapa contoh persamaan pembezaan separa dan keputusan

yang diperolehi telah dibandingkan dengan keputusan yang diperolehi dari kaedah

MLDQ, untuk menunjukkan kejituan kaedah berangka tersebut. Kaedah MLDQ telah

dibangunkan dengan memperbanyakkan bilangan titik grid di kawasan kritikal, dan

juga menganggarkan terbitan pada titik grid tertentu yang dipilih. Kaedah ini

bersama-sama dengan kaedah Runge-Kutta peringkat keempat (RK-4) telah

digunakan dalam persamaan pembezaan seperti persamaan gelombang dan masalah

kecerunan tinggi. Kaedah MLDQ boleh mencapai penyelesaian yang lebih jitu

berbanding dengan kaedah FD yang mempunyai peringkat kejituan yang rendah

dengan menggunakan bilangan titik grid yang kecil. Kaedah multiscale digunakan di

kawasan kritikal kerana mampu memecahkan rantau yang dikehendaki dari titik grid

kasar kepada titik grid lebih perinci. Tambahan lagi, program FORTRAN dengan

kaedah MLDQ telah dibangunkan untuk menyelesaikan masalah-masalah tersebut di

atas. Bagi pengkomputeran selari, seni bina memori perkongsian telah dilaksanakan

dengan menggunakan OpenMP bertujuan untuk mengurangkan masa yang diambil

dalam simulasi keputusan berangka. Dengan itu, keputusan menunjukkan bahawa

kaedah MLDQ adalah teknik berangka yang baik dalam SMA dua dimensi.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xii

LIST OF APPENDICES xvi

1 INTRODUCTION 1 1.1 Background of the Problem 1

1.2 Statement of the Problem 3

1.3 Research Objectives 4

1.4 Scope of the Research 4

1.5 Significance of the Research 5

2 LITERATURE REVIEW 6 2.1 Differential Quadrature Method 6

2.2 Localized Differential Quadrature Method 18

2.3 Multiscale Method 20

2.4 Multiscale Localized Differential 21

Quadrature Method

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2.5 Runge-Kutta Method 21

2.6 Parallel Programming 22

2.6.1 OpenMP 24

2.7 Shape Memory Alloy Problem 26

3 MULTISCALE LOCALIZED

DIFFERENTIAL QUADRATURE METHOD 33

3.1 Introduction 33

3.2 Finite Difference Method 34

3.3 Multiscale Localized Differential 37

Quadrature Method

3.3.1 Critical Region 37

3.3.2 Error Analysis 43

3.4 Sample Applications of MLDQ method 48

3.4.1 Wave Equation 49

3.4.1.1 Results and Analysis 50

3.4.2 Diffusion Equation 56


3.4.3 High Gradient Problem 63


4 PARALLEL PROGRAMMING IN

SOLVING THE BOUNDARY VALUE

PROBLEM 69

4.1 Introduction 69

4.2 Parallel Computing 70

4.3 Results and Analysis 77

4.3.1 Example of Wave Equation 77

4.3.2 Example of Diffusion Equation with

Large Localized Gradient 80

ix

5 MULTISCALE LOCALIZED

DIFFERENTIAL QUADRATURE

APPROACH IN SHAPE MEMORY

ALLOY PROBLEM 84

5.1 Introduction 84

5.2 MLDQ Formulations for SMA problem 85

5.3 Numerical Examples 91

5.3.1 SMA 1 91


5.3.2 SMA 2 96


6 CONCLUSION AND RECOMMENDATION 103

6.1 Conclusion 103

6.2 Recommendation 105

REFERENCES 106

Appendices A-H 112-152

x

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1.1 The Description of a variety of DQ method. 8

2.6.1 The Description of SMA Problems. 27

3.4.1 The measurements of errors when different types

of grid distribution used at time, t=0.6s for

wave equation. 55



diffusion equation. 62



high gradient problem. 68

4.3.1 Table of speedup, S(p) and efficiency, Ep

for different number of grid distribution

for wave equation by using LDQ method. 79

4.3.2 Table of speedup, S(p) and efficiency, Ep

for different number of grid distribution

for diffusion equation by using LDQ method. 81

xi

5.3.1 Numerical results and convergence of

SMA problem with different grid points

for section 5.3.1: SMA 1. 95

xii

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1.1 Integral of u(x) over an interval. 15

2.5.1 The Fork-Join Model. 26

2.6.1 The Different Phases of SMA.( Ozbulut et al. 2011). 32

3.1 Discretization of MLDQ method in cell. 38

3.2 The types of cells in MLDQ method. 38

3.3 Discretization of MLDQ method. 39

3.4 The coordinates in the interval h0

(without multiscale) and h1 (with multiscale). 46

3.5 The coordinates in the interval

h0 (without multiscale) and h1 (with multiscale)

with different total number of interval h0 and h1. 47

3.4.1 Two-dimensional problem with error solutions:

comparison between MLDQ method and

LDQ method solution for wave equation. 52

xiii


comparison between LDQ method and

FD method solution for wave equation. 53

3.4.3 Discretization of MLDQ method in cell for

wave equation. 54

3.4.3 Spatial Convergence with different total number

of grid point for wave equation. 55



LDQ method solution for diffusion equation. 59


comparison between LDQ method and

FD method solution for diffusion equation. 60


Diffusion equation. 61 3.4.8 Spatial Convergence with different total number

of grid point for diffusion equation. 62

3.4.9 Two dimensional problem with error solutions:


LDQ method solution for high gradient problem. 65


high gradient problem. 66

3.4.11 The stability of high gradient problem by using:

(a) LDQ method; (b) MLDQ method. 67

4.2.1 Computation grid with parallelism across space

for LDQ method. 70

xiv

4.2.2 Algorithm for LDQ method. 72

4.2.3 Computation grid with parallelism across space

for MLDQ method. 73

4.2.4 Algorithm for MLDQ method. 75

4.2.5 Serial execution and parallel execution in

OpenMP program. 76

4.3.1 Graph of speedup, S(p) against number of cores,

p by LDQ method for wave equation. 79


p by LDQ method for diffusion equation. 81


p by parallel MLDQ method. 82

5.3.1 Graph of the deviatoric strain, e2 computed by

MLDQ method when t=3s and t=9s for

section 5.3.1: SMA 1. 93

5.3.2 Graph of the temperature, computed by



5.3.3 Graph of the deviatoric strain, e2 and

temperature, computed by FV method

when t=3s and t=9s for section 5.3.1: SMA 1.

(Wang and Melnik, 2007). 95

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5.3.5 Graph of the temperature, computed by



5.3.6 Graph of the deviatoric strain, e2 and

temperature, computed by MLDQ method

when t=2s and t=8s for section 5.3.2: SMA 2.

(Wang and Melnik, 2007) 100



section 5.3.2: SMA 2 when total number of

grid point is 5×5. 101

5.3.8 Discretization of MLDQ method in cell

for section 5.3.2: SMA 2. 102

xvi

LIST OF APPENDICES

APPENDIX TITLE PAGE

A The four approaches of determination of

Weighting Coefficients 112

B Derivations of formulations in Differential

Quadrature Method 116

C Comparison between Finite Difference (FD)

Method and Differential Quadrature (DQ)

Method in One- and two-dimensional

Differential Equation 120

D Comparison between Finite Difference (FD)

Method, Localized Differential Quadrature (LDQ)

Method and Multiscale Localized Differential

Quadrature (MLDQ) method in two-dimensional

Wave Equation 128

E The Shape Memory Effect and the Superelastic

Effect on a Stress-strain Curve 135

F The Truncation Error of the First and Second

Order Derivatives 138

xvii

G The Figures of Thermomchanical waves in SMA 151

H Publications Related to Thesis. 152

1

CHAPTER 1

INTRODUCTION

1.1 Background of Problem

In most of the science and engineering fields, a set of partial differential

equations (PDEs), either linear or nonlinear, the solutions to them must be sorted for.

Therefore, the numerical computations have attracted considerable attention for

solving PDEs problems. There are many available numerical methods used

nowadays for solving PDEs, that is, Finite Difference (FD) method, Finite Element

(FE) method, Finite Volume (FV) method, Boundary Element (BE) method and

others numerical methods. However, many numerical methods have been discussed

and applied in the science and engineering areas, the Differential Quadrature (DQ)

method is by far the most effective tool available to researchers with interests in

numerical computations. The DQ method will be discussed in this study.

The DQ method is more efficient numerical which requires less

computational effort and achieves an acceptable and reasonable accuracy for the

PDEs. Besides, DQ method is an extension of FD method for the higher order of

finite difference scheme. Although DQ method is a numerical technique of high

accuracy, but it is sensitive to the number of grid points. Therefore, many researchers

2

have developed new methods to overcome the limitation of the DQ method. A new

class of numerical methods for solving the sciences and engineering problems will be

discussed in Chapter Two.

Another numerical discretization technique that will be discussed in this

study is Localized Differential Quadrature (LDQ) method. According to Zong and

Lam (2002), LDQ method is characterized by approximating the derivatives at a grid

point using weighted sum of the points in its neighbourhood. This method is used to

solve the limitation of the DQ method. Besides, the Runge-Kutta method has been

discussed in order to numerically integrate the LDQ numerical system in the time

direction.

In this study, parallel programming of shared memory architecture (OpenMP)

is introduced in order to reduce the execution time for sequential algorithm in solving

the PDEs using LDQ method. Generally, when the number of grid points increase

and the simulation time will become longer, this make the whole simulation will

become expensive. Therefore, parallel computation technique will be applied in

solving PDEs.

Furthermore, Multiscale method is also been discussed in this study. The

Multiscale method is a powerful tool for the numerical solution of differential

equation which is based on discrezation and subsequent approximation of derivatives

by FD formulas. The main idea of Multiscale is to accelerate the convergence of base

iterative method by solving a coarse problem. According to Zhu and Cangellaris

(2006), the Multiscale method can be applied in combination with any common

discretization techniques. Based on the multiscale concept with using various

interpolation techniques, in our study, the certain grid point which is out of uniformly

distributed grids can be calculated accurately with less computation time. By take

into account of the powerful Multiscale approach with LDQ method, we calculate the

numerical solution in any point with minimum calculation.

3

Moreover, in this study, we present our method applied in Shape Memory

Alloy (SMA) problems. SMA are novel and special materials which have the ability

to return to predetermined shape when heated above a certain transition temperature.

Constitutive modelling of SMA has been an interest research subject from 1980s

until now. There are many researchers used the various numerical methods in the

numerical simulation of SMA problems. Therefore, we present our method applied in

simple SMA model to achieve a good numerical solution.

1.2 Statement of the Problem

There are many available numerical methods used to approximate the

solution of diffusion equation and wave equation. A set of initial and boundary

conditions are needed to solve these equations. Although the DQ method has been

applied successfully to a variety of science and engineering problems, however, this

method possesses several undesirable limitations and drawbacks. As an example, the

total of grid points used in DQ method is limited due to the ill-conditioned matrix

form. Besides, asymmetry of the final solution matrix produced by the DQ method

makes the solution procedure to be inefficient. Due to overcome this drawback of the

DQ method, many researchers have developed and improved new DQ methods. In

this study, the Multiscale Localized Differential Quadrature (MLDQ) method is

applied in solving the boundary value problems, and also to overcome the limitation

of the DQ method. Finally, parallel programming with shared memory architecture

using OpenMP is implemented in order to reduce the execution time of FORTRAN

program developed in this study.

4

1.3 Objectives of the Study

The objectives of this research are summarized as:

i. To govern MLDQ method in solving wave equation, diffusion equation and

high gradient problem.

ii. To compare the MLDQ method with FD method in terms of their accuracy

and convergence study of numerical solution in solving wave equation,

diffusion equation and high gradient problem.

iii. To govern MLDQ method in the numerical simulation of SMA model.

iv. To develop FORTRAN program codes based on MLDQ method in solving

wave equation, diffusion equation, high gradient problem and SMA problem.

v. To parallelize FORTRAN program codes using OpenMP language for LDQ

method and MLDQ method in solving boundary value problems.

1.4 Scope of the Study

In this research, the basic concept of DQ, LDQ and multiscale methods will

be discussed, and also, understanding these numerical discretization techniques’

application in solving boundary value problems. Another scope of the study will be

focused on solving the two dimensional wave equation and two dimensional high

gradient problem. The Runge-Kutta (RK) method will be utilized in MLDQ method

to numerically integrate it in time direction. Furthermore, FORTRAN program codes

will be developed and parallelized by using share memory architecture, that is,

OpenMP for LDQ method in solving boundary value problems. The limitation of

parallel programming in this study is four cores will be used. In this research, the

multi-core computer Intel® Core™ i5 CPU M460 @2.53GHz is used to do the

programming. The data of the programming is reasonable for four processors to run.

Besides, the MLDQ method will also be implemented in the numerical simulation of

SMA problem.

5

1.5 Significance of the Study

In this research, MLDQ method will be discussed and applied to boundary

value problems. This research is important to overcome the limitations and

drawbacks of the DQ method. Besides, this method also is applied in SMA problem.

Next, the FORTRAN program codes for the LDQ method will be developed in

convenience of checking the performances of the numerical methods. Furthermore,

the OpenMP language is used to parallelize the FORTRAN program codes in order

to reduce the execution time.

106

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